Koosis:(adsbygoogle = window.adsbygoogle || []).push({}); Statistics..., 4th ed., p. 177:

The given answer is yes. "You must be prepared to assumed the populations from which you are sampling are normally distributed" and that "the teaching method is the only reasonable explanation of the differences between groups." A number of students is assigned randomly to three classes with three different teaching methods. The following statistics summarize the performance of the three groups [...] Can you perform an analysis of variance with these data? What assumptions are involved?

Group 1: n = 10, s^{2}= 100.

Group 2: n = 11, s^{2}= 81.

Group 2: n = 8, s^{2}= 64.

In the next problem, the situation is the same, but the data are different. Now the sample variances are 144, 81 and 64. Can ANOVA be done on these data? Answer no, the sample variances are too different.

The obvious question: How different is too different?

One thought I had was to do an F test with null hypothesis the population variances are equal, alternative the population with the biggest sample variance has a bigger population variance than the population with the smallest sample variance. But in Excel, with a 5% significance level, I get a critical value of 3.68. The F scores for both ratios of sample variances are less than this: 100/64 = 1.56 and 144/64 = 2.25. So I guess this isn't the criterion. But why not? And what is?

Another question: what are the populations in this case: three sets each consisting of a hypothetical continuum of infinitely many identical students? Or three copies of the same finite set of actual students, depending on context? Or some ill-defined three copies of the same large, but finite set of all students in history who might conceivably be taught, or have been taught, by these methods, whose population parameters are only approximated by the normal probability measure? Or is it not advisable to think too hard about what population means in such cases?

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# Sample variances & ANOVA: How different is too different?

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